Continuous Histograms for Anisotropy of 2D Symmetric Piece-wise Linear Tensor Fields

TL;DR

This paper develops an analytical method to accurately analyze the anisotropy of 2D tensor fields using continuous histograms, improving the understanding of tensor invariants and their topological features.

Contribution

It introduces a novel analytical derivation of the contour spectrum for quadratic tensor invariants in 2D piece-wise linear tensor fields, enabling precise anisotropy analysis.

Findings

Derived an analytical expression for invariant distribution in tensor fields.

Validated the approach with comparison to naive linear interpolation methods.

Provided a triangulation-based method for accurate contour tree computation.

Abstract

The analysis of contours of scalar fields plays an important role in visualization. For example the contour tree and contour statistics can be used as a means for interaction and filtering or as signatures. In the context of tensor field analysis, such methods are also interesting for the analysis of derived scalar invariants. While there are standard algorithms to compute and analyze contours, they are not directly applicable to tensor invariants when using component-wise tensor interpolation. In this chapter we present an accurate derivation of the contour spectrum for invariants with quadratic behavior computed from two-dimensional piece-wise linear tensor fields. For this work, we are mostly motivated by a consistent treatment of the anisotropy field, which plays an important role as stability measure for tensor field topology. We show that it is possible to derive an analytical expression for the distribution of the invariant values in this setting, which is exemplary given for the anisotropy in all details. Our derivation is based on a topological sub-division of the mesh in triangles that exhibit a monotonic behavior. This triangulation can also directly be used to compute the accurate contour tree with standard algorithms. We compare the results to a na\"ive approach based on linear interpolation on the original mesh or the subdivision.